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Daily structure

The GR4J model (modèle du Génie Rural à 4 paramètres Journalier) is a lumped four-parameter model.

Its four parameters to be calibrated are:

X1 : capacity of the production store  (mm)

X2 : groundwater exchange coefficient  (mm)

X3 : one day ahead capacity of the routing store  (mm)

X4 : time base of unit hydrograph HU1 (d)

In the following, raw rainfall is denoted by P (mm) and potential evapotranspiration (PE) is denoted by E (mm)

P is an estimate of catchment areal rainfall and E can be simply a regime curve of PE repeated every year. The following equation are the results of integrations over the time step.

The first operation is the subtraction of E from P to determine either a net rainfall Pn or a net evapotranspiration capacity En. In GR4J, this operation is computed as if there were an interception storage of zero capacity. Pn and En are computed with the following equations:

Si      P > E,   alors     Pn = P - E        et         En = 0

Si      P < E  alors     Pn = 0              et         En = EP

In case Pn is not zero, a part Ps of Pn fills the production store. It is determined as a function of the level S in the store by:

where X1 (mm) is the maximum capacity of the SMA store.

In the other case, when En is not zero, an actual evaporation rate is determined as a function of the level in the production store to calculate the quantity Es of water that will evaporate from the store. It is obtained by:

The water content in the production store is then updated with:

S = S - Es + Ps

A percolation leakage Perc from the production store is then calculated as a power function of the reservoir content:

Perc is always lower than S. The reservoir content becomes:

The total quantity Pr of water that reaches the routing functions is given by:

Pr = Perc + (Pn – Ps)

Pr is divided into two flow components according to a fixed split: 90 % of Pr is routed by a unit hydrograph UH1 and then a non linear routing store, and the remaining 10 % of Pr are routed by a single unit hydrograph UH2.

Both unit hydrographs depend on the same time parameter X4 expressed in days.

In their discrete form, unit hydrographs UH1 and UH2 have n and m ordinates respectively, where n and m are the smallest integers exceeding x4 and 2.x4 respectively. This means that the water is staggered into n unit hydrograph inputs for UH1 and m inputs for UH2. The ordinates of both unit hydrographs are derived from the corresponding S-curves (cumulative proportion of the input with time) denoted by SH1 and SH2 respectively.

SH1 is defined along time t by:

For t <= 0

For  0 < t < X4

For  t > X4

SH2 is defined along time t by:

For t <= 0

For  0 < t < X4

For  X4 < t < 2X4

For  t > 2X4

UH1 and UH2 ordinates are then calculated by:

where j is an integer.

At each time step i, the outputs Q9 and Q1 from the unit hydrographs are calculated by :

where l = int(X4)+1 and m = int(2.X4)+1, with int(.) the integer part.

A groundwater exchange term F that acts on both flow components, is then calculated as:

where R is the level in the routing store, x3 its “reference” capacity and x2 the water exchange coefficient. x2 can be either positive in case of water imports, negative for water exports or zero when there is no water exchange.

The level in the routing store is updated by adding the output Q9 of UH1 and F as follows:

R = max (0 ; R + Q9 + F)

The outflow Qr of the reservoir is then calculated as:

The level in the reservoir becomes:

R = RQr

Like the content of the routing store, the output Q1 of UH2 is subject to the same water exchange F to give the flow component Qd as follows:

Qd = max (0 ; Q1+F)

Total streamflow Q is finally obtained by:

Q = Qr + Qd

References:

Edijatno, Nascimento, N.O., Yang, X., Makhlouf, Z. et Michel, C. (1999). GR3J : a daily watershed model with three free parameters. Hydrological Sciences Journal, 44(2), 263-278.

Perrin, C., 2002. Vers une amélioration d'un modèle global pluie-débit au travers d'une approche comparative. La Houille Blanche, n°6/7 : 84-91.

Perrin, C., Michel, C. and Andréassian, V., 2003. Improvement of a parsimonious model for streamflow simulation. Journal of Hydrology, 279 : 275-289.

 © Cemagref, mise à jour 14/11/2008